Nuprl Lemma : weighted-sum-split
∀[p,q:ℚ List]. ∀[F:ℕ||p @ q|| ⟶ ℚ].
(weighted-sum(p @ q;F) = (weighted-sum(p;F) + weighted-sum(q;λi.(F (i + ||p||)))) ∈ ℚ)
Proof
Definitions occuring in Statement :
weighted-sum: weighted-sum(p;F),
qadd: r + s,
rationals: ℚ,
length: ||as||,
append: as @ bs,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
weighted-sum: weighted-sum(p;F),
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
uimplies: b supposing a,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
guard: {T},
squash: ↓T,
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
length-append,
sum_split_q,
length_wf,
rationals_wf,
non_neg_length,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
itermAdd_wf,
int_term_value_add_lemma,
qmul_wf,
int_seg_wf,
append_wf,
lelt_wf,
select_wf,
int_seg_properties,
decidable__lt,
add-is-int-iff,
intformless_wf,
int_formula_prop_less_lemma,
false_wf,
qadd_wf,
squash_wf,
true_wf,
qsum_wf,
equal_wf,
select_append_front,
iff_weakening_equal,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
list_wf,
sum_shift_q,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
select_append_back,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
natural_numberEquality,
hypothesisEquality,
addEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
productElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
because_Cache,
applyEquality,
functionExtensionality,
setElimination,
rename,
dependent_set_memberEquality,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
universeEquality,
imageMemberEquality,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
functionEquality,
axiomEquality
Latex:
\mforall{}[p,q:\mBbbQ{} List]. \mforall{}[F:\mBbbN{}||p @ q|| {}\mrightarrow{} \mBbbQ{}].
(weighted-sum(p @ q;F) = (weighted-sum(p;F) + weighted-sum(q;\mlambda{}i.(F (i + ||p||)))))
Date html generated:
2018_05_22-AM-00_34_12
Last ObjectModification:
2017_07_26-PM-06_59_46
Theory : randomness
Home
Index